18 Mr. J. J. Sylvester on a remarkable Modification 



And now suppose that neither q x nor q ni the first or last of 

 the quotients, lies between +1 and — 1, and that no one of the 

 intermediate quotients g 2 , q^ . . . q n ~i lies between -+■ 2 and —2 ; 

 so that, in other words, 



(ft) >* (ft) >* (%) >*> • • • (ft-0 >2 (ft) >x i ' 

 then, I say, that M lf M 2 , M 3 , . . . M„ will have the same signs 

 as q lt ft, ft, ... ft respectively; for 



M 1=!7l , .v(K,)>lj 

 but 



:=ft+^ ••• (M 2 ) = fe)±(^)^2±l; 



•\ M 2 has the same sign as ft, and also (M 2 ) > 1 ; 

 /. in like manner, 



(M 8 ) has the same sign as ft, and also (M 3 ) > 1 ; 

 /.in like manner, 



(M 4 ) has the same sign as ft, and also (M 4 ) >■ 1 ; 



and so on until we come to M (n _u, and we shall find 



M n _i of the same sigh as ft_i, and also (M w _!) > 1. 

 Finally, 



M 2 



where (ft) > 1 and (jp- J 



<1, 



.\ M n has the same sign as (ft) ; 

 but we cannot say (nor is there any occasion to say) that (M„)> 1 ; 

 ,\ D = Mj . M g . M 8 . . . M n has the same sign as ft . ft . q 3 . . . q n . 



Now let f(x) be any given function of x of the rath degree, 

 and <f>(x) any assumed function whatever of x of the (n — l)th 

 degree, and let 



£# = _1_ J }_ 1 



fa ft + ft + ft + " &* 

 where q l9 q^ ft . . . q n are now supposed to be linear functions 

 of x, which, except for special relations between /and <f>, will 

 always exist, and can be found by the ordinary process of suc- 

 cessive division. 



Write down the n pairs of equations, 

 v 1 = <7 1 + l=0 M 2 s=y 2 + 2 = w 3 =</3 + 2=0...ii n =sgr n -fl = o, 

 u\ = qi -l=0 w' 2 =<7 2 -2 = u f 3 =q 3 -2 = 0...u! n =q n -l = 0. 



If the greatest of the values of x determined from these 2n 

 equations be called L, and the last of these values be called A, 



